Dini's surface

In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere.^[1] It is named after Ulisse Dini^[2] and described by the following parametric equations:^[3]

${\displaystyle {\begin{aligned}x&=a\cos u\sin v\\y&=a\sin u\sin v\\z&=a\left(\cos v+\ln \tan {\frac {v}{2}}\right)+bu\end{aligned}}}$

Dini's surface with 0 ≤ u ≤ 4π and 0.01 ≤ v ≤ 1 and constants a = 1.0 and b = 0.2.

Dini's surface plotted with adjustable parameters by Wolfram Mathematica program

Dini's Surface with constants a = 1, b = 0.5 and 0 ≤ u ≤ 4π and 0<v<1.

Another description is a generalized helicoid constructed from the tractrix.^[4]

See also

• Breather surface

References

1. "Wolfram Mathworld: Dini's Surface". Retrieved 2009-11-12.
2. J J O'Connor and E F Robertson (2000). "Ulisse Dini Biography". School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on 2012-06-09. Retrieved 2016-04-12.
3. "Knol: Dini's Surface (geometry)". Archived from the original on 2011-07-23. Retrieved 2009-11-12.
4. Rogers and Schief (2002). Bäcklund and Darboux transformations: geometry and modern applications in Soliton Theory. Cambridge University Press. pp. 35–36.